{"id":29791,"date":"2026-10-27T11:23:14","date_gmt":"2026-10-27T11:23:14","guid":{"rendered":"https:\/\/blog.oratrics.com\/?p=29791"},"modified":"2026-10-28T05:51:10","modified_gmt":"2026-10-28T05:51:10","slug":"square-root-by-division-method","status":"publish","type":"post","link":"https:\/\/oratrics.com\/blogs\/math-explorer-program\/square-root-by-division-method\/","title":{"rendered":"How to find square root of any number by long division method"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"29791\" class=\"elementor elementor-29791\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cc2aee8 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"cc2aee8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-27af257\" data-id=\"27af257\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-678bf2f\" data-id=\"678bf2f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-83e3ddc elementor-widget elementor-widget-heading\" data-id=\"83e3ddc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">How to Find the Square Root of Any Number by the Long Division Method<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9cfef53 elementor-widget elementor-widget-text-editor\" data-id=\"9cfef53\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Some square roots are easy. You see 49 and you just know it\u2019s 7. Then a test paper throws 5476 or 71 at you, and suddenly guessing doesn\u2019t cut it.<\/span><\/p><p><span style=\"font-weight: 400;\">That\u2019s when the <\/span><b>square root by division method<\/b><span style=\"font-weight: 400;\"> earns its place. It works on any number: huge, tiny, perfect square, not-so-perfect square, even decimals. Is it slower than a calculator? Yes. But after a few rounds of practice it starts to feel almost automatic, because you\u2019re not being clever, you\u2019re just repeating the same little loop until the answer appears.<\/span><\/p><p><span style=\"font-weight: 400;\">By the end of this guide you\u2019ll know how the method works and why, how to run it step by step, and how to handle perfect squares, non-perfect squares and decimals. You\u2019ll also know the slip-ups that quietly cost students marks, and how to check your work in about ten seconds.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3c4f6db elementor-align-center elementor-widget elementor-widget-button\" data-id=\"3c4f6db\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/demo.oratrics.com\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Book a Free Demo Today<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-42f3b8d elementor-widget elementor-widget-image\" data-id=\"42f3b8d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Joyful-Square-Roots-Classroom-1024x576.webp\" class=\"attachment-large size-large wp-image-29875\" alt=\"How to find square root of any number by long division method\" srcset=\"https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Joyful-Square-Roots-Classroom-1024x576.webp 1024w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Joyful-Square-Roots-Classroom-300x169.webp 300w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Joyful-Square-Roots-Classroom-768x432.webp 768w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Joyful-Square-Roots-Classroom-1536x864.webp 1536w, https:\/\/blog.oratrics.com\/wp-content\/uploads\/2026\/10\/Joyful-Square-Roots-Classroom.webp 1672w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-abbe7a9\" data-id=\"abbe7a9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-981864e elementor-widget elementor-widget-html\" data-id=\"981864e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t\t<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\r\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Poppins:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\r\n\r\n<div class=\"toc-wrapper\">\r\n    \r\n    \r\n    <div class=\"custom-toc\" id=\"custom-toc\">\r\n        <div class=\"toc-title\">\r\n            \u2630 Table of Contents\r\n        <\/div>\r\n        <ul class=\"toc-list\" id=\"toc-list\"><\/ul>\r\n    <\/div>\r\n    \r\n<\/div>\r\n\r\n<style>\r\n\/* Parent Fixed Wrapper holding both components together *\/\r\n.toc-wrapper {\r\n    position: fixed;\r\n    top: 150px; 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\r\n        overflow-y: visible;\r\n    }\r\n\r\n    .custom-toc .toc-title {\r\n        font-size: 25px;\r\n    }\r\n\r\n    .custom-toc .toc-list a {\r\n        font-size: 14px;\r\n    }\r\n}\r\n<\/style>\r\n\r\n<script>\r\ndocument.addEventListener(\"DOMContentLoaded\", function () {\r\n    const tocList = document.getElementById(\"toc-list\");\r\n    const h1 = document.querySelector(\"h1\");\r\n    const h2s = document.querySelectorAll(\"h2\");\r\n\r\n    function createId(text) {\r\n        return text.toLowerCase().replace(\/[^a-z0-9]+\/g, '-');\r\n    }\r\n\r\n    if (h1) {\r\n        if (!h1.id) {\r\n            h1.id = createId(h1.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.innerHTML = `<a href=\"#${h1.id}\">${h1.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    }\r\n\r\n    h2s.forEach(h2 => {\r\n        if (!h2.id) {\r\n            h2.id = createId(h2.innerText);\r\n        }\r\n        const li = document.createElement(\"li\");\r\n        li.classList.add(\"toc-h2\");\r\n        li.innerHTML = `<a href=\"#${h2.id}\">${h2.innerText}<\/a>`;\r\n        tocList.appendChild(li);\r\n    });\r\n});\r\n<\/script>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4cfc41e elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"4cfc41e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-07f63a0\" data-id=\"07f63a0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-d9d571f\" data-id=\"d9d571f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-db70554 elementor-widget elementor-widget-heading\" data-id=\"db70554\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">What is the square root by division method?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-365bf25 elementor-widget elementor-widget-text-editor\" data-id=\"365bf25\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Quick refresher first. The square root of a number is whatever you multiply by itself to get that number back. Since 12 \u00d7 12 = 144, we say \u221a144 = 12.<\/span><\/p><p><span style=\"font-weight: 400;\">The long division method (you\u2019ll also see it called the <\/span><i><span style=\"font-weight: 400;\">division method<\/span><\/i><span style=\"font-weight: 400;\"> or the <\/span><i><span style=\"font-weight: 400;\">digit-by-digit method<\/span><\/i><span style=\"font-weight: 400;\">) builds that answer one digit at a time. Each round you find the biggest digit that still fits, subtract, bring down the next chunk of the number, and go again.<\/span><\/p><p><span style=\"font-weight: 400;\">Why bother with it when other methods exist?<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Prime factorisation is lovely for perfect squares, but it gets you nowhere with a number like 71.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Estimation gets you close, but \u201cclose\u201d doesn\u2019t earn full marks.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Long division works on everything, and you can keep going to as many decimal places as you like.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">If your textbook is covering squares, radicals or number systems, this is almost certainly the method it expects for bigger numbers.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-812fce2\" data-id=\"812fce2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c6afb20 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"c6afb20\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-59d6ab4\" data-id=\"59d6ab4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-9476b5b\" data-id=\"9476b5b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dee74d1 elementor-widget elementor-widget-heading\" data-id=\"dee74d1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">A few words you\u2019ll see along the way<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-288af4e elementor-widget elementor-widget-text-editor\" data-id=\"288af4e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Don\u2019t worry about memorising these. Just skim them so nothing feels foreign later.<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Radicand:<\/b><span style=\"font-weight: 400;\"> the number sitting under the root sign. In \u221a529, it\u2019s 529.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Perfect square:<\/b><span style=\"font-weight: 400;\"> a whole number that\u2019s the square of another whole number (1, 4, 9, 16, 25, 36&#8230;).<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Non-perfect square:<\/b><span style=\"font-weight: 400;\"> a number like 2, 71 or 128. Its square root is a decimal that never ends and never repeats.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Pair (or period):<\/b><span style=\"font-weight: 400;\"> a group of two digits. The method chops your number into these.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Divisor and quotient:<\/b><span style=\"font-weight: 400;\"> same meaning as in ordinary long division. The quotient is the answer you\u2019re building up.<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-db8f070\" data-id=\"db8f070\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d773072 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"d773072\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8935379\" data-id=\"8935379\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-2339aa0\" data-id=\"2339aa0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-68c24b2 elementor-widget elementor-widget-heading\" data-id=\"68c24b2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">The method, step by step<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4dba568 elementor-widget elementor-widget-text-editor\" data-id=\"4dba568\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Here\u2019s the whole recipe. Read it once, then see it in action in the examples, because it makes a lot more sense when there are real numbers involved.<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Pair up the digits.<\/b><span style=\"font-weight: 400;\"> Start at the decimal point and work outwards. For a whole number, that means pairing from the right, so if there\u2019s an odd number of digits, the leftmost group will be a lone digit. For decimals, pair the digits after the point from left to right, and add a zero if the last pair is incomplete.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Deal with the first group.<\/b><span style=\"font-weight: 400;\"> Find the biggest number whose square fits into it (is less than or equal to it). That\u2019s the first digit of your answer, and also your first divisor.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Subtract.<\/b><span style=\"font-weight: 400;\"> Multiply, subtract, and you\u2019ve got a remainder.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Bring down the next pair<\/b><span style=\"font-weight: 400;\"> and stick it next to the remainder. That\u2019s your new dividend.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Double the answer so far.<\/b><span style=\"font-weight: 400;\"> Write it down with a blank beside it, like 4_.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Fill in the blank.<\/b><span style=\"font-weight: 400;\"> Pick the biggest digit that makes (your number with that digit) \u00d7 (that digit) fit into the dividend.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Subtract, then repeat.<\/b><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Stop<\/b><span style=\"font-weight: 400;\"> when you hit a remainder of zero (a perfect square) or when you\u2019ve got as many decimal places as you need.<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">If you want one line to remember: <\/span><b>divide, multiply, subtract, bring down, repeat.<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-59c72a3\" data-id=\"59c72a3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ef37988 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"ef37988\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7253934\" data-id=\"7253934\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-bfd14e9\" data-id=\"bfd14e9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2570ece elementor-widget elementor-widget-heading\" data-id=\"2570ece\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Example 1: A perfect square (529)<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a9ee84d elementor-widget elementor-widget-text-editor\" data-id=\"a9ee84d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><b>Pair the digits:<\/b><span style=\"font-weight: 400;\"> 529 becomes <\/span><b>5 | 29<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>First group:<\/b><span style=\"font-weight: 400;\"> the biggest square that fits into 5 is 4 (2 \u00d7 2). So the first digit of the answer is <\/span><b>2<\/b><span style=\"font-weight: 400;\">, and 5 \u2212 4 leaves <\/span><b>1<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Bring down 29<\/b><span style=\"font-weight: 400;\"> and you\u2019ve got <\/span><b>129<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Double the answer so far:<\/b><span style=\"font-weight: 400;\"> 2 doubled is 4, so the new divisor is <\/span><b>4_<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Find the digit:<\/b><span style=\"font-weight: 400;\"> try 3. 43 \u00d7 3 = 129, which is exactly what we need. The answer is now <\/span><b>23<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Subtract:<\/b><span style=\"font-weight: 400;\"> 129 \u2212 129 = <\/span><b>0<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">Nothing left over, so <\/span><b>\u221a529 = 23<\/b><span style=\"font-weight: 400;\">. Done.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ea32287\" data-id=\"ea32287\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-16d881a elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"16d881a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-aa407b5\" data-id=\"aa407b5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-85dc5c0\" data-id=\"85dc5c0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2a2bba4 elementor-widget elementor-widget-heading\" data-id=\"2a2bba4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Example 2: A bigger number (7056)\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a5be48c elementor-widget elementor-widget-text-editor\" data-id=\"a5be48c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Pair it up: <\/span><b>70 | 56<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The biggest square that fits into 70 is 64 (8 \u00d7 8). Answer so far: <\/span><b>8<\/b><span style=\"font-weight: 400;\">. Remainder: 70 \u2212 64 = <\/span><b>6<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Bring down 56 to get <\/span><b>656<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Double 8 to get 16, so the divisor is <\/span><b>16_<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">164 \u00d7 4 = 656. Answer: <\/span><b>84<\/b><span style=\"font-weight: 400;\">. Remainder: <\/span><b>0<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">So <\/span><b>\u221a7056 = 84<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">Notice that a four-digit number needed only two rounds. That\u2019s not a coincidence: the number of pairs always matches the number of digits in your answer.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-9a65a93\" data-id=\"9a65a93\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0adeb60 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"0adeb60\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8fdd246\" data-id=\"8fdd246\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-8ad58ad\" data-id=\"8ad58ad\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ce329e3 elementor-widget elementor-widget-heading\" data-id=\"ce329e3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Example 3: A non-perfect square (71)\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c1682e7 elementor-widget elementor-widget-text-editor\" data-id=\"c1682e7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This is where the method really shows its worth. Since 71 isn\u2019t a perfect square, the remainder won\u2019t ever settle at zero, so we\u2019ll stop at three decimal places.<\/span><\/p><p><span style=\"font-weight: 400;\">Write 71 as <\/span><b>71 . 00 00 00<\/b><span style=\"font-weight: 400;\">, adding three pairs of zeros for three decimal places.<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The biggest square that fits into 71 is 64. Answer so far: <\/span><b>8<\/b><span style=\"font-weight: 400;\">. Remainder: <\/span><b>7<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Put the decimal point in the answer. Bring down 00 to get <\/span><b>700<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Double 8 to get 16, so the divisor is <\/span><b>16_<\/b><span style=\"font-weight: 400;\">. Try 4: 164 \u00d7 4 = 656. Remainder: <\/span><b>44<\/b><span style=\"font-weight: 400;\">. Answer: <\/span><b>8.4<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Bring down 00 to get <\/span><b>4400<\/b><span style=\"font-weight: 400;\">. Double 84 to get 168, so the divisor is <\/span><b>168_<\/b><span style=\"font-weight: 400;\">. Try 2: 1682 \u00d7 2 = 3364. Remainder: <\/span><b>1036<\/b><span style=\"font-weight: 400;\">. Answer: <\/span><b>8.42<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Bring down 00 to get <\/span><b>103600<\/b><span style=\"font-weight: 400;\">. Double 842 to get 1684, so the divisor is <\/span><b>1684_<\/b><span style=\"font-weight: 400;\">. Try 6: 16846 \u00d7 6 = 101076. Remainder: <\/span><b>2524<\/b><span style=\"font-weight: 400;\">. Answer: <\/span><b>8.426<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">So <\/span><b>\u221a71 \u2248 8.426<\/b><span style=\"font-weight: 400;\">, correct to three decimal places.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-5ab244e\" data-id=\"5ab244e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c563c12 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"c563c12\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7b6bba3\" data-id=\"7b6bba3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-33315ae\" data-id=\"33315ae\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9f36d76 elementor-widget elementor-widget-heading\" data-id=\"9f36d76\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Example 4: A decimal number (12.25)\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e18b08e elementor-widget elementor-widget-text-editor\" data-id=\"e18b08e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">With decimals, pair the digits on each side of the point separately: <\/span><b>12 . 25<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The biggest square that fits into 12 is 9. Answer so far: <\/span><b>3<\/b><span style=\"font-weight: 400;\">. Remainder: <\/span><b>3<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Put the decimal point in the answer and bring down 25 to get <\/span><b>325<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Double 3 to get 6, so the divisor is <\/span><b>6_<\/b><span style=\"font-weight: 400;\">. Try 5: 65 \u00d7 5 = 325. Remainder: <\/span><b>0<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">So <\/span><b>\u221a12.25 = 3.5<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">Here\u2019s a neat way to sanity-check decimal answers: the root has half as many decimal places as the original number. 12.25 has two, and 3.5 has one.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-536a8c4\" data-id=\"536a8c4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fb16338 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"fb16338\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-618e447\" data-id=\"618e447\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a5ea3a6\" data-id=\"a5ea3a6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8227cb2 elementor-widget elementor-widget-heading\" data-id=\"8227cb2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Example 5: The square root of 2\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-aad1784 elementor-widget elementor-widget-text-editor\" data-id=\"aad1784\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This one turns up <\/span><i><span style=\"font-weight: 400;\">everywhere<\/span><\/i><span style=\"font-weight: 400;\">, so it\u2019s worth getting comfortable with. Write 2 as <\/span><b>2 . 00 00 00<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The biggest square that fits into 2 is 1. Answer: <\/span><b>1<\/b><span style=\"font-weight: 400;\">. Remainder: <\/span><b>1<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Bring down 00 to get <\/span><b>100<\/b><span style=\"font-weight: 400;\">. Double 1 to get 2, so the divisor is <\/span><b>2_<\/b><span style=\"font-weight: 400;\">. Try 4: 24 \u00d7 4 = 96. Remainder: <\/span><b>4<\/b><span style=\"font-weight: 400;\">. Answer: <\/span><b>1.4<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Bring down 00 to get <\/span><b>400<\/b><span style=\"font-weight: 400;\">. Double 14 to get 28, so the divisor is <\/span><b>28_<\/b><span style=\"font-weight: 400;\">. Try 1: 281 \u00d7 1 = 281. Remainder: <\/span><b>119<\/b><span style=\"font-weight: 400;\">. Answer: <\/span><b>1.41<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Bring down 00 to get <\/span><b>11900<\/b><span style=\"font-weight: 400;\">. Double 141 to get 282, so the divisor is <\/span><b>282_<\/b><span style=\"font-weight: 400;\">. Try 4: 2824 \u00d7 4 = 11296. Remainder: <\/span><b>604<\/b><span style=\"font-weight: 400;\">. Answer: <\/span><b>1.414<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">So <\/span><b>\u221a2 \u2248 1.414<\/b><span style=\"font-weight: 400;\">. You could keep going all day and the digits would never fall into a repeating pattern, which is one of the signs that \u221a2 is irrational.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-42f3995\" data-id=\"42f3995\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-24ec801 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"24ec801\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3f6ac21\" data-id=\"3f6ac21\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-127d537\" data-id=\"127d537\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d8e5671 elementor-widget elementor-widget-heading\" data-id=\"d8e5671\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Example 6: Finding the least number to subtract<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1027067 elementor-widget elementor-widget-text-editor\" data-id=\"1027067\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Exam setters love this type of question. <\/span><b>What\u2019s the smallest number you must subtract from 1500 to get a perfect square?<\/b><\/p><p><span style=\"font-weight: 400;\">Run the division on 1500, paired as <\/span><b>15 | 00<\/b><span style=\"font-weight: 400;\">:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The biggest square that fits into 15 is 9. Answer so far: <\/span><b>3<\/b><span style=\"font-weight: 400;\">. Remainder: <\/span><b>6<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Bring down 00 to get <\/span><b>600<\/b><span style=\"font-weight: 400;\">. Double 3 to get 6, so the divisor is <\/span><b>6_<\/b><span style=\"font-weight: 400;\">. Try 8: 68 \u00d7 8 = 544. Remainder: <\/span><b>56<\/b><span style=\"font-weight: 400;\">. Answer: <\/span><b>38<\/b><span style=\"font-weight: 400;\">.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">The remainder is 56, which means 38\u00b2 sits exactly 56 below 1500. Subtract it:<\/span><\/p><p><span style=\"font-weight: 400;\">1500 \u2212 56 = <\/span><b>1444<\/b><span style=\"font-weight: 400;\">, and \u221a1444 = <\/span><b>38<\/b><span style=\"font-weight: 400;\">.<\/span><\/p><p><b>Answer: 56.<\/b><\/p><p><span style=\"font-weight: 400;\">What if the question asked what to <\/span><i><span style=\"font-weight: 400;\">add<\/span><\/i><span style=\"font-weight: 400;\"> instead? Then you\u2019d head for the next perfect square. 39\u00b2 = 1521, and 1521 \u2212 1500 = <\/span><b>21<\/b><span style=\"font-weight: 400;\">.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-a990837\" data-id=\"a990837\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-646ed42 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"646ed42\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c8023ab\" data-id=\"c8023ab\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-7d1fddc\" data-id=\"7d1fddc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4d9bad1 elementor-widget elementor-widget-heading\" data-id=\"4d9bad1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">A few shortcuts that actually help\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-73a564e elementor-widget elementor-widget-text-editor\" data-id=\"73a564e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Know your squares up to about 20.<\/b><span style=\"font-weight: 400;\"> It makes the first step of almost every problem instant.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Peek at the last digit.<\/b><span style=\"font-weight: 400;\"> A perfect square can only end in 0, 1, 4, 5, 6 or 9. If your number ends in 2, 3, 7 or 8, don\u2019t expect a whole-number answer.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Count the pairs.<\/b><span style=\"font-weight: 400;\"> The number of pairs tells you how many digits the whole-number part of the root will have.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Keep your columns neat.<\/b><span style=\"font-weight: 400;\"> Honestly, most mistakes come from messy layout, not from the maths.<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-ab9b9b5\" data-id=\"ab9b9b5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-dead6fa elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"dead6fa\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-518f77a\" data-id=\"518f77a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-96fce9c\" data-id=\"96fce9c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6133c37 elementor-widget elementor-widget-heading\" data-id=\"6133c37\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Mistakes worth avoiding\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e3d46cf elementor-widget elementor-widget-text-editor\" data-id=\"e3d46cf\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Pairing from the wrong side.<\/b><span style=\"font-weight: 400;\"> Whole numbers are paired from the right. Decimals are paired outwards from the decimal point, in both directions.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Doubling only the last digit.<\/b><span style=\"font-weight: 400;\"> You double the <\/span><i><span style=\"font-weight: 400;\">whole<\/span><\/i><span style=\"font-weight: 400;\"> answer so far, not just the digit you found last.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Picking a digit that\u2019s too big.<\/b><span style=\"font-weight: 400;\"> If your product overshoots the dividend, drop the digit by one and try again.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Bringing down single zeros.<\/b><span style=\"font-weight: 400;\"> When you add zeros for decimal places, bring them down in pairs.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Putting the decimal point in the wrong place.<\/b><span style=\"font-weight: 400;\"> It goes into the answer the moment you bring down your first decimal pair.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Skipping the check.<\/b><span style=\"font-weight: 400;\"> Ten seconds at the end can save you a lost mark, as you\u2019re about to see.<\/span><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4237221\" data-id=\"4237221\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b797bc3 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"b797bc3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-c436bf5\" data-id=\"c436bf5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-f9c5671\" data-id=\"f9c5671\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-867232e elementor-widget elementor-widget-heading\" data-id=\"867232e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">How to check your answer\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1f83a6d elementor-widget elementor-widget-text-editor\" data-id=\"1f83a6d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">For a perfect square, multiply your answer by itself and see if you get the original number back. 23 \u00d7 23 = 529, so we\u2019re good.<\/span><\/p><p><span style=\"font-weight: 400;\">For a non-perfect square, square your approximation and see how close it lands. 8.426\u00b2 comes out at about 70.997, which is very nearly 71. Carry the division further and it gets closer still.<\/span><\/p><p>Read More:<\/p><ol><li><p id=\"application-for-leave-of-absence-for-offices-school-and-college\" class=\"elementor-heading-title elementor-size-default\"><strong><a href=\"https:\/\/oratrics.com\/blogs\/english-concept\/application-for-leave\/\">Application for Leave of Absence<\/a><\/strong><\/p><\/li><li><p id=\"132000-in-words-spelling-place-value-chart-easy-examples\" class=\"elementor-heading-title elementor-size-default\"><strong><a href=\"https:\/\/oratrics.com\/blogs\/maths\/132000-in-words\/\">132000 in Words<\/a><\/strong><\/p><\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e7d7de7\" data-id=\"e7d7de7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-664d689 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"664d689\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-39b634d\" data-id=\"39b634d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-01e68cb\" data-id=\"01e68cb\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-593eab2 elementor-widget elementor-widget-heading\" data-id=\"593eab2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">When should you actually use this method?\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-aaab2bc elementor-widget elementor-widget-text-editor\" data-id=\"aaab2bc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<table><tbody><tr><td><p><b>Situation<\/b><\/p><\/td><td><p><b>Best method<\/b><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Small perfect squares (up to about 400)<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Memorised squares or prime factorisation<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Large perfect squares<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Long division<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Non-perfect squares<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Long division<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Decimal numbers<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Long division<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">A quick rough answer<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">Estimating between two perfect squares<\/span><\/p><\/td><\/tr><tr><td><p><span style=\"font-weight: 400;\">Everyday life, no rules<\/span><\/p><\/td><td><p><span style=\"font-weight: 400;\">A calculator, obviously<\/span><\/p><\/td><\/tr><\/tbody><\/table><p><span style=\"font-weight: 400;\">Calculators are great outside the classroom. But in plenty of school exams, especially in the middle and early high school years, you\u2019re expected to show your working. That\u2019s exactly where this method comes in.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-bc11bb2\" data-id=\"bc11bb2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a008a01 elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-id=\"a008a01\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-77ba09d\" data-id=\"77ba09d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-df9a51b\" data-id=\"df9a51b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3cca7e0 elementor-widget elementor-widget-heading\" data-id=\"3cca7e0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Frequently Asked Questions\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7c36c99 elementor-widget elementor-widget-accordion\" data-id=\"7c36c99\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1301\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1301\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">What is the square root by division method?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1301\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1301\"><p><span style=\"font-weight: 400;\">\u00a0It\u2019s a digit-by-digit technique for finding a square root. You pair up the digits, then repeatedly divide, multiply, subtract and bring down the next pair.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1302\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-1302\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Why do we group digits in pairs?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1302\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-1302\"><p><span style=\"font-weight: 400;\">Because squaring a one-digit number gives at most two digits, and squaring a two-digit number gives at most four. So every pair of digits in the number lines up with exactly one digit in the root.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1303\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-1303\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><svg class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><svg class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">What about non-perfect squares?<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1303\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-1303\"><p><span style=\"font-weight: 400;\">Add pairs of zeros after the decimal point and keep going until you\u2019ve got the accuracy you need. The remainder will never reach zero.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"What is the square root by division method?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">\\u00a0It\\u2019s a digit-by-digit technique for finding a square root. You pair up the digits, then repeatedly divide, multiply, subtract and bring down the next pair.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"Why do we group digits in pairs?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">Because squaring a one-digit number gives at most two digits, and squaring a two-digit number gives at most four. So every pair of digits in the number lines up with exactly one digit in the root.<\\\/span><\\\/p>\"}},{\"@type\":\"Question\",\"name\":\"What about non-perfect squares?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<p><span style=\\\"font-weight: 400;\\\">Add pairs of zeros after the decimal point and keep going until you\\u2019ve got the accuracy you need. The remainder will never reach zero.<\\\/span><\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3331a1d\" data-id=\"3331a1d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>How to Find the Square Root of Any Number by the Long Division Method Some square roots are easy. You [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":29875,"comment_status":"open","ping_status":"open","sticky":false,"template":"rttpg_full_width","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[9],"tags":[],"class_list":["post-29791","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math-explorer-program"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - 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